The point of anesthesia is to not feel anything during an operation, so if an anesthetic is working successful, you will not feel anything. Some forms of anesthesia however will leave the patient still aware of what is going on during the operation. IV monitored anesthesia is reduced so that the patient remains conscious. This may be if it is best during the operation that a patient remains conscious. During this process, the patient may be aware of what is happening but will not feel anything.
Regional anesthesia numbs a part of the body so that part of the body does not feel anything. The rest of the body however is still responsive to stimuli and the patient remains conscious. With local anesthesia, it will be the same situation where the person will be aware and only the area around where the operation is being done will be numbed. Although you may be aware of what is going on under anesthesia, if it is working properly, you will not feel anything.